Mission-critical systems, such as unmanned aerial vehicles (UAVs), chemical reactors, and paper mills, are required to operate continuously for a period to complete a mission. During operation, unexpected system failure is possible. When there is a sign of imminent failures, the mission can be aborted to increase the system survival probability and minimize damage to the system. We consider the abort decision-making problem for systems with three states, i.e., healthy, defective, and failure, where the healthy and defective states are unobservable before failure. Condition-monitoring sensors are installed on the system and periodically generate signals indicating the system health state. Mission abort can be made based on these signals. However, possible measurement errors and environmental noises result in imperfect sensitivity and specificity of the sensors, which significantly complicates the decision-making. Furthermore, the random time from system defect to failure typically has an increasing hazard rate, leading to a non-Markovian transition between the defective and failure state. This study meets these challenges by adopting the Erlang mixture distributions to approximate the non-Markovian failure process as a continuous-time Markov chain in a new state space. A partially observable Markov decision process (POMDP) is then formulated for decision-making. We show that the optimal policy of the POMDP follows a control-limit structure in a spherical coordinate system, and the sequence of optimal policies converges to the true optimal policy for the original problem when the phase of the Erlang mixture goes to infinity. A modified point-based value iteration algorithm is developed to deal with the curse of dimensionality. We further investigate two special cases of our model that can be exactly solved after discretizing the state space. Through a case study on a UAV, we demonstrate the capability of real-time implementation of our model, even when the condition-monitoring signals are generated with high frequency.